abstract: In the lectures we will illustrate some of the results of Kolmogorov, Arnol'd, Moser, Herman, Yoccoz and others on stability of quasiperiodic motions for various kinds of dynamical systems. We will emphasize the interplay between analytical assumptions and the number-theoretical conditions needed for the proofs. We will also possibly address some of the most recent developments of the theory, including the problem of uniqueness of KAM curves and the linearization of perturbations of linear flows on higher genus surfaces.